Semihypergroups obtained by merging of 0-semigroups with groups

Abstract

We consider the class of 0-semigroups (H,ӿ) that are obtained by adding a zero element to a group (G, ·) so that for all x, y ∈ G it holds x ӿ y ≠0  ⇒    x ӿ y = xy. These semigroups are called 0-extensions of (G, ·). We introduce a merging operation that constructs a 0-semihypergroup from a 0-extension of (G, ·) by a suitable superposition of the product tables. We characterize a class of 0-simple semihypergroups that are merging of a 0-extension of an elementary Abelian 2-group. Moreover, we prove that in the finite case all such 0-semihypergroups can be obtained from a special construction where  (H,ӿ) is nilpotent